Non-smooth time decomposition for nonlinear models driven by random pulses

In this work, the time argument is decomposed according to impact interactions in a chain of absolutely rigid and perfectly elastic particles. Such a decomposition leads to explicit equations of the stroboscopic mapping for a general class of dynamical systems. Between the times of observation, the system motion is approximated by the Lie series. For illustration, Duffingʹs oscillator with no linear stiffness under the sine-modulated sequence of Diracʹs pulses, that is a modified Uedaʹs model, was considered. In some cases, a slight randomization of the pulse times could significantly suppress the mapping chaos which is caused by the system nonlinearity. Monte Carlo simulation showed also that such a small random irregularity of the input can bring out the system orbits more clearly in stroboscopic phase plots of the dynamics. An asymmetric Van der Pol equation under the regular and random pulse sequences was considered as another example which is related to the nerve pulse propagation modeling adopted in Biology. In some cases, irregularity of the pulse times resulted in more organized structures of the stroboscopic diagrams.